Bernstein Functions and Radial Limits of Prescribed Mean Curvature Surfaces

Abstract

The radial limits at a point ${\bf y}$ of the boundary of the domain $\Omega\subset {\bf R}^{2}$ of a bounded variational solution $f$ of Dirichlet or contact angle boundary value problems for a prescribed mean curvature equation are studied with an emphasis on the effects of assumptions about the curvatures of the boundary $\partial\Omega$ on each side of the point ${\bf y}.$ For example, at a nonconvex corner ${\bf y},$ we previously proved that all nontangential radial limits of $f$ at ${\bf y}$ exist, here we provide sufficient conditions for the tangential radial limits to exist, even when the Dirichlet data $\phi\in L^{\infty}(\partial\Omega)$ has no one-sided limits at ${\bf y}$ or the contact angle $\gamma\in L^{\infty}(\partial\Omega:[0,\pi])$ is not bounded away from $0$ or $\pi.$ We also provide a complement to a 1976 Theorem by Leon Simon on least area surfaces.